熵作为拓扑运算导子
Entropy as a Topological Operad Derivation
摘要：
我们在信息论、代数和拓扑学之间有一个小小的联系--也就是说，香农熵和拓扑单形运算的导子之间有一个对应关系。我们首先简要回顾了运算及其表示与拓扑单纯形和真正的线作为主要的例子。然后，我们给出了一个一般的定义，在任何类别的一个操作数的衍生物的值在一个交换双模的操作。主要结果是香农熵定义了拓扑单形的操作数的导数，并且对于这个操作数的每一个导数，都存在一个点，在该点上它由香农熵的常数倍给出。我们表明，这是兼容的，并在很大程度上依赖于，一个众所周知的表征熵由Faddeev在1956年和最近的变化Leinster。
英文摘要：
We share a small connection between information theory, algebra, and topology - namely, a correspondence between Shannon entropy and derivations of the operad of topological simplices. We begin with a brief review of operads and their representations with topological simplices and the real line as the main example. We then give a general definition for a derivation of an operad in any category with values in an abelian bimodule over the operad. The main result is that Shannon entropy defines a derivation of the operad of topological simplices, and that for every derivation of this operad there exists a point at which it is given by a constant multiple of Shannon entropy. We show this is compatible with, and relies heavily on, a well-known characterization of entropy given by Faddeev in 1956 and a recent variation given by Leinster.
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